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TobyBartels comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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Yes, and I was happy to take it this way, as I am certainly no Platonist. Surely only a Platonist could believe that AC is true; we philosophically sophisticated people know that you can make whatever assumptions you want! And so naturally a theorem with a proof using AC is a weaker result than the same theorem with a proof that doesn't, since it holds under fewer sets of assumptions, and thus the latter is preferred. Meanwhile, a theorem with a proof using not(AC) is just as valid as the same theorem with a proof using AC; it's less useful only because it has fewer connections with the published corpus of mathematics, but that's merely a sociological contingency.
Is it often the case that you need to assume the negation of AC for a proof to hold? AC comes up in seemingly-unrelated areas when you need some infinitely-hard-to-construct object to exist; I can't imagine a similar case where you'd assume not(AC) in, e.g., ring theory.
As usual, the negation of a useful statement ends up not being a useful statement. I don't think anyone works with not(AC), they work with various stronger things that imply not(AC) but actually have interesting consequences.
That's intriguing. Do you have any examples of what people actually work with?
Sniffnoy may have more examples, but here are some that I know:
Adding the first two to ZF + DC (dependent choice) is consistent (assuming that ZFC + Con(ZFC) is consistent, as just about everybody believes), and this gives a "dream universe" for analysis in which, for example, any everywhere-defined linear operator between Hilbert spaces is bounded.
This isn't quite right. The consistency of ZF + DC + "every subset of R is Lebesgue measurable" is equivalent to the consistency of an inaccessible cardinal, which is a much stronger assumption then the consistency of ZFC + Con(ZFC).
Sorry, my mistake. Still, set theorists usually believe this.
Yes, indeed!
Yes -- but it needs to be stressed that this doesn't distinguish AC from anything else! (Also, depending on the context, there may other criteria for selecting proofs besides the strength or weakness of their assumptions.)
If only people would talk about whether they prefer working in ZFC or ZF+not(C) (or plain ZF), or better yet what they like and don't like about each, rather than whether AC is "true" or how "skeptical" they are.
Yes, indeed, that would be much more sophisticated! But scepticism of the orthodoxy can be the first step to such sophistication. (It was for me, although in my case there were also some parallel first steps that did not initially seem connected.)